What are partial fractions for and how do I find them?

Being able to find partial fractions is very useful for complex algebraic fractions that cannot be intergrated in their current form. Breaking it down into a sum of it's partial fractions gives two (or more) simplier fractions that can be easily intergrated. To find partial fractions, first check if the fraction is improper (top-heavy). If it is, perform algebraic division to give a non-fraction and a proper algeraic fraction, of which the partial fractions can now be found. Once this is done, or if the fraction was orginally proper, factorise the denominator. Make the original fraction equal to the sum of an unknown constant over one factor and an unknown constant over the other factor (e.g. A/factor1 + B/factor2), these are the partial fractions. To find the constants, add these fractions together and you will see that this new numerator is equal to numerator of the original fraction. This can be done by setting x to a value so one of the factors equals zero, so that one constant can be found. Now set the other factor to zero so that the other constant can be found. You now have the two constants for the numerators so can form the full partial fractions. A similar method can be used if the original denominator has 3 factors, with the last constant being found by comparing coeffiecients. If there is a repeated root, this part will produce two partial fractions.